Extremal functions for an embedding from some anisotropic space, involving the 'one Laplacian'

Françoise Demengel; Thomas Dumas

doi:10.3934/dcds.2019048

Article Contents

2019, Volume 39, Issue 2: 1135-1155. Doi: 10.3934/dcds.2019048

This issue Previous Article Mass concentration phenomenon to the 2D Cauchy problem of the compressible Navier-Stokes equations Next Article Characterization for the existence of bounded solutions to elliptic equations

Extremal functions for an embedding from some anisotropic space, involving the "one Laplacian"

Françoise Demengel^, and
Thomas Dumas

UMR 8088, University of Cergy Pontoise, 2 avenue Adolphe Chauvain, Cergy, France

* Corresponding author: Françoise Demengel
* Corresponding author: Françoise Demengel

Received: March 31, 2018

Revised: June 30, 2018

Published: January 2019

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Abstract

In this paper, we prove the existence of extremal functions for the best constant of embedding from anisotropic space, allowing some of the Sobolev exponents to be equal to $1$. We prove also that the extremal functions satisfy a partial differential equation involving the $1$ Laplacian.

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Mathematics Subject Classification: Primary: 35L30; Secondary: 35J75, 35J60.

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